Properties of the Fourier Transform
CS/BIOEN 4640: Image Processing Basics
March 22, 2012
Review: Euler’s Representation
Im
r
φ
0
Re
I
A complex number can be
given as an angle φ and a
radius r
I
Think 2D polar coordinates
I
Exponential form:
reiφ = r cos(φ) + i (r sin(φ))
Review: Fourier Transform
Given a complex-valued function g : R → C,
Fourier transform produces a function of frequency ω :
Z ∞
h
i
1
G(ω) = √
g(x) · cos(ωx) − i · sin(ωx) dx
2π Z−∞
∞
1
g(x) · e−iωx dx
=√
2π −∞
Inverse Fourier Transform
The Fourier transform is invertible. That is, given the
Fourier transform G(ω) we can reconstruct the original
function g as
1
g(x) = √
2π
Z
∞
G(ω) · eiωx dω
−∞
We use the notation:
Fourier transform:
G = F{g}
Inverse Fourier transform: g = F −1 {G}
The Dirac Delta
Definition
The Dirac delta or impulse is defined as
Z
δ(x) = 0 for x 6= 0,
∞
δ(x) dx = 1
and
−∞
I
The Dirac delta is not a function
I
It is undefined at x = 0.
I
Has the property
Z
∞
f (x) δ(x) dx = f (0)
−∞
for any function f
The Dirac Delta
1.0
0.5
0.0
δ(x)
1.5
2.0
Even though the Dirac delta is not a function, we will plot
it like this:
−1.0
−0.5
0.0
x
0.5
1.0
2.0
1.5
1.0
2δδ(x)
0.0
0.5
1.0
0.5
0.0
δ(x − 0.5)
1.5
2.0
Shifting and Scaling Deltas
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
x
A shifted impulse is
given by δ(x − a)
0.0
0.5
1.0
x
A scaled impulse is
given by s · δ(x)
R∞
−∞ s
· δ(x) dx = s
Fourier Transform Pairs: Cosine
1.0
1.5
1.0
0.5
0.5
−5
−3
−1
1
3
5
−5
−3
−1
1
3
5
−0.5
−0.5
−1.0
−1.0
g(x) = cos(ω0 x)
(Here ω0 = 3)
−1.5
G(ω) =
pπ
2
· (δ(ω + ω0 ) + δ(ω − ω0 ))
Fourier Transform Pairs: Sine
1.0
1.5i
1.0i
0.5
0.5i
−5
−3
−1
1
3
5
−5
−3
−1
1
3
5
−0.5i
−0.5
−1.0i
−1.0
g(x) = sin(ω0 x)
(Here ω0 = 5)
−1.5i
G(ω) = i
pπ
2
· (δ(ω + ω0 ) − δ(ω − ω0 ))
BOOK TYPO
Fourier Transform Pairs: Gaussian
−9
−7
−5
−3
1.0
1.0
0.5
0.5
−1
1
3
5
7
9
−9
−7
−5
−3
−1
−0.5
−0.5
−1.0
−1.0
g(x) =
1
σ
exp
−x2
2σ 2
(Here σ = 3)
G(ω) = exp
1
3
5
−ω 2
2·(1/σ 2 )
7
9
Fourier Transform Pairs: Box
−9
−7
−5
−3
1.5
1.5
1.0
1.0
0.5
0.5
−1
1
3
5
7
9
−0.5
(
1
g(x) =
0
−9
−7
−5
−3
−1
1
3
5
−0.5
if |x| < b
otherwise
(Here b = 2)
G(ω) =
2 sin(b ω)
√
2π ω
BOOK TYPO
7
9
Properties: Linearity
Scaling:
F{c · g(x)} = c · G(ω)
Addition:
F{g1 (x) + g2 (x)} = G1 (ω) + G2 (ω)
Properties for Real-Valued Functions
I
If g is a real-valued function, then
G(ω) = G∗ (−ω)
I
I
If g is real-valued and even: g(x) = g(−x), then
G(ω) is real-valued and even
If g is real-valued and odd: g(x) = −g(−x), then
G(ω) is purely imaginary and odd
Properties: Similarity
Stretching a signal horizontally leads to a shrinking of
the Fourier spectrum:
ω 1
F{g(s · x)} =
·G
|s|
s
And vice versa, shrinking the signal causes a stretching
in the Fourier spectrum
Properties: Shift
A horizontal shift of the signal results in a phase shift of
the Fourier transform:
F{g(x + d)} = e−iωd · G(ω)
I
Notice magnitude |G(ω)| stays the same
I
This is a rotation in the complex plane by −ωd
Properties: Convolution
Convolution becomes multiplication in the Fourier
domain:
F{g(x) ∗ h(x)} =
√
2π G(ω) · H(ω)
And vice versa, multiplication becomes convolution:
1
F{g(x) · h(x)} = √ G(ω) ∗ H(ω)
2π